Principle Bundles and Hopf Map for Spheres

Principle bundles and Hopf map for spheres

Irina Markina, University of Bergen, Norway

Summer school Analysis - with Applications to Mathematical Physics Gottingen¨ August 29 - September 2, 2011

Principle bundles and Hopf map for spheres – p. 1/46 Fiber bundle

The smooth ﬁber bundle is a collection of smooth manifolds F,M,B and a smooth map π : M → B.

∀ x ∈ M, there is U(π(x)) ⊂ B, such that π−1(U) is diffeomorphic to the product space U × F :

φ M ⊃ π−1(U) / U × F, oo ooo π oo oo pr woo 1 B ⊃ U

∀ b ∈ B, the pre-image π−1(b) is diffeomorphic to F and is called the ﬁber over b. B is the base space, M is the total space and the map π is the projection map.

Principle bundles and Hopf map for spheres – p. 2/46 Fiber bundle

Sea urchin

Principle bundles and Hopf map for spheres – p. 3/46 Vertical space

Fiber bundle π : M → B

dqπ : TqM → Tπ(q)B is surjective ⇒ ker(dqπ) = ∅

Denote ker(dqπ) = Vq and call the vertical space.

The collection of all vertical space is vertical distribution or vertical sub-bundle V ⊂ TM.

Vq = Tq(F ) for ﬁber F passing through q

Principle bundles and Hopf map for spheres – p. 4/46 Ehresmann connection

An Ehresmann connection for π : M → B is a distribution D ⊂ TM that is everywhere transverse and of complementary dimension to V :

TqM = Dq ⊕ Vq. q ∈ M.

⊕ denotes only transversality, but not orthogonality.

Dp is a horizontal vector space and the Ehresmann connection D is a horizontal distribution on M.

Given π : M → B we have V = ker(dπ). Construction of D is the question of a mathematical art.

Principle bundles and Hopf map for spheres – p. 5/46 Sub-Riemannian structure by restriction

Let π :(M,gM ) → B, Vq = ker(dqπ)

Deﬁne Dq ⊕⊥ Vq = TqM. The distribution D is the Ehresmann connection

Let gD := gM |D

(M,D,gD) is the sub-Riemannian manifold deﬁned by submersion π and the Riemannian metric gM . The sub-Riemannian length of a horizontal curve is equal to the Riemannian length, since the vertical components vanish.

Principle bundles and Hopf map for spheres – p. 6/46 Induced sub-Riemannian structure

Let π : M → (B,gB) and D is deﬁned.

Pullback the metric gB to the horizontal distribution D:

gD(v,w) := gB(dqπ(v),dqπ(w)), v,w ∈ Dq, q ∈ M.

(M,D,gD) is sub-Riemannian manifold induced by the Ehresmann connection D and the Riemannian metric gB. If γ is horizontal in M, then

lsR(γ) = lgB (π(γ)).

The vertical component of γ˙ is absent

Principle bundles and Hopf map for spheres – p. 7/46 Induced sub-Riemannian structure

The horizontal lifting of c: I → B is a curve γ : I → M such that

γ˙ (t) ∈ Dγ(t) and π(γ(t)) = c(t) for all t ∈ I.

The horizontal lift of a Riemannian geodesic in B is a sub-Riemannian geodesic in M.

If the Riemannian geodesic in B is a length minimizers between b0,b1 ∈ B, then its horizontal lift is a sub-Riemannian length minimizer between the corresponding ﬁbers.

Principle bundles and Hopf map for spheres – p. 8/46 When two structures coincide?

Suppose π :(M,gM ) → (B,gB), gMD := gM |D. and D is given. If the restriction

dqπ|Dq :(Dq,gMD ) → (Tπ(q)B,gB) is a linear isometry for all q ∈ M, then

(M,D,gMD )=(M,D,gBD ).

The submersion π is called the Riemannian submersion.

Principle bundles and Hopf map for spheres – p. 9/46 Principal bundle

A smooth ﬁber bundle (F,M,B,π) is a principal bundle if a ﬁber F has the structure of a Lie group G and

: F × G → F q.τ → rτ (q) = qτ acts (on the right) freely and transitively on each ﬁber.

G acts freely on the right on M if

∀ q ∈ M, q.τ = q.ς if and only if τ = ς.

G acts transitively on the right on M if

∀ q,p ∈ M there is τ ∈ G such that q.τ = p

Principle bundles and Hopf map for spheres – p. 10/46 Metric on principal bundles

Let π :(M,gM ) → B, Vq = ker(dqπ), D ⊕⊥ V and let gM be G-right invariant:

gM (vq,wq) = gM drτ (vq), drτ (wq) , τ ∈ G Then

gD = gM |D is the sub-Riemannian metric

gV = gM |V is the vertical metric

Fibers are isomorphic to a Lie groups G, so gV can be encode in the Lie algebra g.

Principle bundles and Hopf map for spheres – p. 11/46 Inertia tensor

The inﬁnitesimal generator σq of the right G-action on M d σq : g → Vq ⊂ TqM such that g ∋ ξ → q. exp(ǫξ). dǫǫ=0 d d σq(ξ) = q. exp(ǫξ) = lq exp(ǫξ) = dlq(ξ) dǫǫ=0 dǫǫ=0 Deﬁne the bilinear symmetric tensor Iq by

Iq(ξ,η) = gV (σq(ξ),σq(η)), ξ,η ∈ g q ∈ M.

Iq is the inertia tensor on g. If Iq is independent on q, then gV is also left invariant.

Principle bundles and Hopf map for spheres – p. 12/46 Metric we are working with

Let π :(M,gM ) → B, Vq = ker(dqπ), D ⊕⊥ V

A Riemannian metric gM is of constant bi-invariant type if

1. gM is right G invariant,

2. its inertia tensor Iq is independent of q ∈ M. Metric we are working with

Let π :(M,gM ) → B, Vq = ker(dqπ), D ⊕⊥ V

A Riemannian metric gM is of constant bi-invariant type if

1. gM is right G invariant,

2. its inertia tensor Iq is independent of q ∈ M.

σq . y . o proj. g l =∼ ker(dqπ) = Vq o TqM −1 σq

A is g-valued connection form

Principle bundles and Hopf map for spheres – p. 13/46 Normal geodesics

Let expR be the Riemannian exponential map for gM ,

γv,R(t) = expR(tv) be the Riemannian geodesic passing through q ∈ M with the initial velocity vector v ∈ TqM.

Lift horizontaly π(γv,R) from B to M and get a curve γsR.

The curve γsR is a normal sub-Riemannian geodesic, given by γsR(t) = γv,R(t)expG(−tA(v)). All normal sub-Riemannian geodesics are given by this formula

Principle bundles and Hopf map for spheres – p. 14/46 Proof of the theorem

gM , gD, gV =⇒ HR,HsR,HV

expR : TM → M, expsR : D → M, expV : V → M. If Hamiltonian functions Poisson commute, then the ﬂows on T ∗M also commute.

If v = vD + vV is the initial velocity vector and the exponential maps commute

expsR(tvD) = expR(tv)expV (−tvV ). The last step is to observe

expV (−tvV ) = expG(−tA(vV )) since the metric gV is bi-invariant. Principle bundles and Hopf map for spheres – p. 15/46 Proof of the theorem

∗ ∗ {HV ,HsR} = 0, ((τ,) × (b,p)) ∈ T G × T U, U ∈ B

HsR = HsR(,b,p),HV = HV ()

l k ∂HsR ∂HV ∂HsR ∂HV {HsR,HV } = + ∂j ∂τj ∂pj ∂bj j=1 j=1

l k ∂H ∂H ∂H ∂H − sR V + sR V = 0. ∂τj ∂j ∂bj ∂pj j=1 j=1

Principle bundles and Hopf map for spheres – p. 16/46 Hopf map

Let S2n+1 = {z ∈ Cn+1 | z2 = 1} right U(1)-action on S2n+1 is

(z0,...,zn).υ =(z0υ,...,znυ), υ ∈ U(1)

It induces the principal U(1)-bundle

h U(1) → S2n+1 −→ CP n

(z0,...,zn) → h(z0,...,zn)=[z0 : : zn], where [z0 : : zn] denotes homogeneous coordinates.

j j zj = x + iy

Principle bundles and Hopf map for spheres – p. 17/46 Odd dimensional spheres

Any sphere S2n+1 possesses the vector ﬁeld

0 0 n n V˜q = −y ∂x0 + x ∂y0 − ... − y ∂xn + x ∂yn .

n+1 that is V˜q = iNq, where i is the usual i in C and 2n+1 n+1 Nq = q is normal to sphere S ⊂ C .

V˜q ∈ ker(dqπ), Vq = span{V˜q}, D ⊕⊥ V, It is also D = ker ω for the contact form

0 0 n n ω = −y0dx + x0dy − ... − yndx + xndy

Principle bundles and Hopf map for spheres – p. 18/46 Geodesics on S2n+1

2n+1 2n+1 S =(S ,D,gD) is the sub-Riemannian manifold

The Lie algebra u(1) = spanR(i)

The u(1)−valued connection form is A(v) = ig(v,Vq), 2n+1 v ∈ TqS

The Riemannian metric g on S2n+1 is of constant bi-invariant type, since for iα ∈ u(1)

d σq(iα) = q exp (ǫ(iα)) = q.iα = αVq, dǫ u(1) ǫ=0 Iq(iα, iα˜) = g(αVq, αV˜ q) = αα˜ does not depend on q ∈ M

Principle bundles and Hopf map for spheres – p. 19/46 Geodesics on S2n+1

2n+1 2n+1 Let q ∈ S and v ∈ TqS . If v γ (t)=(z (t),...,z (t)) = q cos(vt) + sin(vt) R 0 n v where v = g(v,v), is the great circle satisfying γR(0) = q and γ˙R(0) = v, then the sub-Riemannian geodesic γsR is given by

−itg(v,Vq) −itg(v,Vq) γsR(t) = z0(t)e ,...,zn(t)e , t ∈ R.

Principle bundles and Hopf map for spheres – p. 20/46 Corollaries

2 2 2 • γ˙sR(t) + g (v,Vq) = v .

γ˙sR is constant and length of γ is 2 2 ℓsR(γ)=(b − a) v − g(v,Vq) for t ∈ [a, b] .

Principle bundles and Hopf map for spheres – p. 21/46 Corollaries

2 2 2 • γ˙sR(t) + g (v,Vq) = v .

γ˙sR is constant and length of γ is 2 2 ℓsR(γ)=(b − a) v − g(v,Vq) for t ∈ [a, b] . • If γsR(t) = γR(t)expu(1)(−itg(v,Vq)) is parameterized 2 2 by arc length then v = 1 + g(v,Vq) .

Principle bundles and Hopf map for spheres – p. 21/46 Corollaries

2 2 2 • γ˙sR(t) + g (v,Vq) = v .

γ˙sR is constant and length of γ is 2 2 ℓsR(γ)=(b − a) v − g(v,Vq) for t ∈ [a, b] . • If γsR(t) = γR(t)expu(1)(−itg(v,Vq)) is parameterized 2 2 by arc length then v = 1 + g(v,Vq) .

• If γ˙R(0) ∈ D then the set of such geodesics is diffeomorphic to CP n.

γ˙R(0) ∈ D ⇒ g(v,Vq) = 0, v thus γsR(t) = q cos(vt) + v sin(vt) is the great circle, whose loci is uniquely determined by the point [v] ∈ CP n.

Principle bundles and Hopf map for spheres – p. 21/46 Parallalizable spheres

n n THEOREM (Adams, 1962) Let S −1 = {x ∈ R : x2 = 1}. Then Sn−1 has precisely ̺(n) − 1 globally deﬁned, linearly independent, non vanishing vector ﬁelds

̺(n) = 2c + 8d, n = (2a + 1)2b, b = c + 4d, 0 ≤ c ≤ 3. In particular, • The even dimensional spheres have no globally defined and non vanishing vector fields, • S1, S3 and S7 are the only spheres with maximal number of linearly independent globally defined non vanishing vector fields.

Principle bundles and Hopf map for spheres – p. 22/46 S3 as a group

3 4 2 2 2 2 S = {x ∈ R : x0 + x1 + x2 + x3 = 1}

Set z1 = x0 + ix1, z2 = x2 + ix3

z1 z2 2 2 SU(2) : , z1,z2 ∈ C, |z1| + |z2| = 1. −z¯2 z¯1

−1 (z1,z2) = (¯z1, −z2), (1, 0) is the unit U(1, H) is the group of unit quaternions, Sp(1) is the special symplectic group, Spin(3) is the spin group on three generators. All they are double cover of the group SO(3).

Principle bundles and Hopf map for spheres – p. 23/46 Right invariant vector ﬁelds

1 x0 x1 x2 x3 x0  0   −x1 x0 −x3 x2   x1  drq(e1) = =  0   −x2 x3 x0 −x1   x2         0   −x3 −x2 x1 x0   x3        3 N = x0∂0 + x1∂1 + x2∂2 + x3∂3 is normal vector to S , g(N,N) = 1,

V = −x1∂0 + x0∂1 − x3∂2 + x2∂3, g(V,V ) = 1, g(V,N) = 0,

X = −x2∂0 + x3∂1 + x0∂2 − x1∂3, g(X,X) = 1, g(X,N) = 0,

Y = −x3∂0 − x2∂1 + x1∂2 + x0∂3, g(Y, Y ) = 1, g(Y,N) = 0, [X,V ] = Y , [V, Y ] = X, [Y,X] = V

Principle bundles and Hopf map for spheres – p. 24/46 Horizontality condition

3 TpS = span{X,Y,V }, D = span{X, Y }. The geometry obtained by ﬁxing other pair of vector ﬁelds is similar. Let γ =(x0,x1,x2,x3) be a curve on S3. Then

γ˙ =x ˙ 0∂0 +x ˙ 1∂1 +x ˙ 2∂2 +x ˙ 3∂3 = a(s)X(γ(s)) + b(s)Y (γ(s)) + c(s)V (γ(s)).

The curve γ is horizontal iff

1 0 3 2 c = g(˙γ,V ) = −x x˙ 0 + x x˙ 1 − x x˙ 2 + x x˙ 3 = 0.

Principle bundles and Hopf map for spheres – p. 25/46 Horizontality condition

1 1 (x dx1 − x dx0) = (x dx3 − x dx2) 2 0 1 2 2 3

(x0,x1)

A01 c

0 A23 (x2,x3)

Projections of c to the planes (x0,x1) and (x2,x3)

Principle bundles and Hopf map for spheres – p. 26/46 CR-structure on S3

3 2 S := {(z1,z2) ∈ C | z1z¯1 + z2z¯2 = 1}. For a point q ∈ M, the holomorphic tangent space of a real manifold M at q is the vector space

HqM = TqM ∩ J(TqM), where J2 = −id is an almost complex structure. A real submanifold M of Cn is said to have a CR structure if dimR HqM does not depend on q ∈ M.

3 HqS is a complex vector space of complex dim. one.

Principle bundles and Hopf map for spheres – p. 27/46 CR-structure on S3

Consider the differential form

ω =z ¯1dz1 +z ¯2dz2, d(z1z¯1 + z2z¯2) Then 3 HqS = ker ω = span{(−X + iY )}.

J(∂xj ) = ∂yj , J(∂yj ) = −∂xj , j = 0, 2.

J(X) = Y, J(Y ) = −X

Principle bundles and Hopf map for spheres – p. 28/46 CR-structure on S3

The same almost complex structure can be obtained by means of the covariant derivative of S3 considered as a smooth Riemannian manifold embedded in R4.

Since ∇X V = Y and ∇Y V = −X one can introduce the operator

J(X) = ∇X V, where ∇ is the Levi-Civitá connection.

A. HURTADO, C.ROSALES, Area-stationary surfaces inside the sub-Riemannian three-sphere. Math. Ann. 340 (2008), no. 3, 675–708.

Principle bundles and Hopf map for spheres – p. 29/46 Hopf map h : S3 → S2 ∼= CP 1 ∼= S3/U(1) can be written in coordinates

2 2 2 h(z1,z2)=(|z1| − |z2| , 2z1z¯2) ∈ S

The action of U(1) on S3 is

2πit 2πit 2πit 2πit 3 (z1,z2).e =(z1e ,z2e ), e ∈ U(1), (z1,z2) ∈ S

2πit 3 φ(t)=(ˆz1, zˆ2)e is a ﬁber over (ˆz1, zˆ2) ∈ S that 2 collapses to h(ˆz1, zˆ2) ∈ S under the Hopf map. The curve φ(t) is a great circle on S3.

˙ 3 ker(dh) = span{φ(t)} = span{V }, ker(dh) ⊕⊥ D = T S

Principle bundles and Hopf map for spheres – p. 30/46 Hopf ﬁbration

The Hopf ﬁbration can be visualized using a stereographic projection of S3 into R3.

iπξ1 iπξ2 z1 = |z1|e , z2 = |z2|e , |z1| = const

Principle bundles and Hopf map for spheres – p. 31/46 Hopf ﬁbration

Here keyrings mimick part of the Hopf ﬁbration by showing some of the circles of the Hopf ﬁbration which lie on the common torus.

Principle bundles and Hopf map for spheres – p. 32/46 Closed geodesics

2n+1 Let γsR : R → S be a complete sub-Riemannian geodesic parameterized by arc-length, with initial 2n+1 velocity v ∈ TqS . Then γsR is closed if and only if

g(v,Vq) ∈ Q. 2 1 + g (v,Vq)

Principle bundles and Hopf map for spheres – p. 33/46 Conclusion

• There are three possible ways to construct the sub-Riemannian structure on S3

Principle bundles and Hopf map for spheres – p. 34/46 Conclusion

• There are three possible ways to construct the sub-Riemannian structure on S3

Action of the group S3 ֒→ R4 of unit quaternions •

Principle bundles and Hopf map for spheres – p. 34/46 Conclusion

• There are three possible ways to construct the sub-Riemannian structure on S3

Action of the group S3 ֒→ R4 of unit quaternions •

CR-structure on hypersurface S3 ֒→ C2 •

Principle bundles and Hopf map for spheres – p. 34/46 Conclusion

• There are three possible ways to construct the sub-Riemannian structure on S3

Action of the group S3 ֒→ R4 of unit quaternions •

CR-structure on hypersurface S3 ֒→ C2 •

h Hopf ﬁbration U(1) ֒→ S3 −→ S2 •

Principle bundles and Hopf map for spheres – p. 34/46 Conclusion

• There are three possible ways to construct the sub-Riemannian structure on S3

Action of the group S3 ֒→ R4 of unit quaternions •

CR-structure on hypersurface S3 ֒→ C2 •

h Hopf ﬁbration U(1) ֒→ S3 −→ S2 •

• Locally the sub-Riemannian structure coincides with the Heisenberg group H1 according to the Gromov-Margulis-Mitchell-Mostow construction of the tangent cone.

Principle bundles and Hopf map for spheres – p. 34/46 Problems with S7